1. 數值方法 2008, Applied Mathematics NDHU1  An iterative approach for root finding  Newton method Lecture 3II Root Finding

2. 數值方法 2008, Applied Mathematics NDHU2 Problem statement Find x such that f(x)=0 for a given f f denotes an arbitrary single-variable function Root finding

3. 數值方法 2008, Applied Mathematics NDHU3 >> x=linspace(-1,3); >> plot(x,2.^x.^2-10*x+1)

4. 數值方法 2008, Applied Mathematics NDHU4 Root finding Input: an expression Plot given function Find a root

5. 數值方法 2008, Applied Mathematics NDHU5 Tangent line x xnxn y=f(x) y n =f(x n ) x n+1

6. Taylor series 數值方法 2008, Applied Mathematics NDHU6

7. 7 An iterative approach Start at some guess Refine current guess by Find a tangent line that passes (x n,f(x n )) Set x n to a point at intersection of the tangent line to horizontal axis

8. 數值方法 2008, Applied Mathematics NDHU8 Updating rule

9. 數值方法 2008, Applied Mathematics NDHU9 Newton method An Iterative approach Random guess Refine current guess by an updating rule If a halting condition holds, go to step 2 otherwise exit

10. 數值方法 2008, Applied Mathematics NDHU10 An iterative approach n=0 ; x n n=0 ; Set x n x n+1 Determine x n+1 Increase n by one Halting condition T F

11. 數值方法 2008, Applied Mathematics NDHU11 An iterative approach x randomly Set x randomly an updating rule Apply an updating rule x to refine x ~( halting condition) exit

12. 數值方法 2008, Applied Mathematics NDHU12 Halting condition Let x denote the current guess The absolute value of f(x) is expected as close as possible to zero Halting Condition abs(f(x)) < epslon epslon denotes a predefined small positive number

13. 數值方法 2008, Applied Mathematics NDHU13 Initialization Random initialization A guess by the user

14. 數值方法 2008, Applied Mathematics NDHU14 An iterative approach x = rand*2-1; s=sym(ss); ep=10^-6 f=inline(s); s1=diff(s); f1=inline(s1) x=x-f(x)/f1(x); fprintf('%f\n',x) ~( abs(f(x)) < ep) exit input an expression, ss

15. 數值方法 2008, Applied Mathematics NDHU15 Demo_newton source codesource code demo_newton fstr=input('input a function:','s'); x_ini=input('guess its zero:'); x_zero=newton(fstr,x_ini); newton.m

16. 數值方法 2008, Applied Mathematics NDHU16 Main Program

17. 數值方法 2008, Applied Mathematics NDHU17 Newton method

18. 數值方法 2008, Applied Mathematics NDHU18 >> demo_newton input a function:cos(x) guess its zero:2 iter=1 x=1.542342 fx=0.028450 iter=2 x=1.570804 fx=-0.000008 iter=3 x=1.570796 fx=0.000000 >>

19. 數值方法 2008, Applied Mathematics NDHU19 Example demo_newton input a function:2.^x.^2-10*x+1 guess its zero:1.5

20. 數值方法 2008, Applied Mathematics NDHU20 Example >> demo_newton input a function:2.^x.^2-10*x+1 guess its zero:0.5

21. Second order expansion 數值方法 2008, Applied Mathematics NDHU21

22. Second order method Update rule 數值方法 2008, Applied Mathematics NDHU22

23. 數值方法 2008, Applied Mathematics NDHU23 An iterative approach x = rand*2-1; s=sym(ss); ep=10^-6 f=inline(s); s1=diff(s); f1=inline(s1) s2=diff(s1); f2=inline(s2) a=f2(x)/2;b=-f2(x)+f1(x);c=f(x)-x*f1(x)+f2(x)*x^2/2;x=… fprintf('%f\n',x) ~( abs(f(x)) < ep) exit input an expression, ss

24. 數值方法 2008, Applied Mathematics NDHU24 Unconstrained Optimization Problem statement Given a differentiable function, y=g(x), unconstrained optimization aims to find the minimum of g(x) Let x denote a minimum and

25. 數值方法 2008, Applied Mathematics NDHU25 Optimization by Newton method Use symbolic differentiation to find the first derivative of a given function Apply the Newton method to find zeros of the first derivative

26. 數值方法 2008, Applied Mathematics NDHU26 An iterative approach x = rand*2-1; s=sym(ss); ep=10^-6 g=inline(s); s1=diff(s); f=inline(s1); s2=diff(s1); f1=inline(s2) x=x-f(x)/f1(x); fprintf('%f gx=%f\n',x, g(x)) ~( abs(f(x)) < ep) exit input an expression, ss

27. 數值方法 2008, Applied Mathematics NDHU27 First derivative ss=input('input a function:','s'); s=sym(ss); f=inline(s); s1=diff(s); f1=inline(s1); x=linspace(-5,5); plot(x,f(x));hold on; plot(x,f1(x));

28. 數值方法 2008, Applied Mathematics NDHU28 >> x=linspace(-2*pi,2*pi); >> plot(x,(x-tanh(2*x+10)).^2)

29. 數值方法 2008, Applied Mathematics NDHU29 >> demo_min input a function:(x-tanh(2*x+10)).^2 guess its minimum:4 Local minimum

30. 數值方法 2008, Applied Mathematics NDHU30 >> demo_min input a function:(x-tanh(2*x+10)).^2 guess its minimum:-4 Local Minimum

31. 數值方法 2008, Applied Mathematics NDHU31 Global minimum

32. 數值方法 2008, Applied Mathematics NDHU32 >> demo_min input a function:0.2*x.^3-3*x+cos(x) guess its minimum:2

33. 數值方法 2008, Applied Mathematics NDHU33 Exercise ex3II.pdf